Which expression is equivalent to \(4\mathrm{x}(3\mathrm{y} - 1) - 7(3\mathrm{y} - 1)\)? \((3\mathrm{y} - 1)(4\mathrm{x} - 7)\) \((3\mathrm{y} - 1)(...

1. INFER the factoring opportunity

• Looking at \(\mathrm{4x(3y - 1) - 7(3y - 1)}\), notice that both terms contain the exact same factor: \(\mathrm{(3y - 1)}\)
• This means we can factor out this common binomial factor

2. SIMPLIFY by applying reverse distributive property

• Factor out \(\mathrm{(3y - 1)}\) from both terms:
  • From \(\mathrm{4x(3y - 1)}\), what's left is \(\mathrm{4x}\)
  • From \(\mathrm{-7(3y - 1)}\), what's left is \(\mathrm{-7}\)
• Result: \(\mathrm{(3y - 1)(4x - 7)}\)

3. Verify the factorization

• Expand \(\mathrm{(3y - 1)(4x - 7)}\) to check:
• \(\mathrm{(3y - 1)(4x - 7) = 3y(4x - 7) - 1(4x - 7) = 12xy - 21y - 4x + 7}\)
• Original: \(\mathrm{4x(3y - 1) - 7(3y - 1) = 12xy - 4x - 21y + 7}\) ✓

Answer: A. \(\mathrm{(3y - 1)(4x - 7)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{(3y - 1)}\) is a common factor that can be factored out. Instead, they try to expand everything first, leading to a more complex expression like \(\mathrm{12xy - 4x - 21y + 7}\), then struggle to factor this four-term polynomial. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the factoring opportunity but make sign errors, particularly with the \(\mathrm{-7}\) term. They might write \(\mathrm{(3y - 1)(4x + 7)}\) instead of \(\mathrm{(3y - 1)(4x - 7)}\), forgetting that \(\mathrm{-7(3y - 1)}\) means the coefficient is \(\mathrm{-7}\), not \(\mathrm{+7}\). This may lead them to select Choice B (\(\mathrm{(3y - 1)(4x + 7)}\)).

The Bottom Line:

Success on this problem requires recognizing patterns - specifically that when the same binomial appears in multiple terms, it can be factored out just like a single variable. The key insight is seeing \(\mathrm{(3y - 1)}\) as a single "unit" that appears in both terms.